Vol 8, No 12 (2017)

RESULTS ON

-NUMBER FOR THE GENERALIZED PETERSEN GRAPHS

P n k

(

,

)

B. JOHN*

1

_{, J. JOSELINE MANORA}

2

_{AND I. PAULRAJ JAYASIMMON}

3

1

Department of Mathematics,

A.J.C. English School, Kumbakonam, Tamil Nadu, India.

2

_{PG & Research Department of Mathematics,}

T.B.M.L College, Porayar. Nagapattinam Dt, India.

3

_{Department of Mathematics,}

Amet University, Kanathur, Chennai, India.

(Received On: 02-11-17; Revised & Accepted On: 24-11-17)

ABSTRACT

A

set S of vertices of a graph G is said to be a Majority Independent set(or MI-set) if it induces a totally disconnected

subgraph with

[ ]

2

p

N S

≥

 

_{ }

 

and

[ , ]

[ ]

2

p

pn v S

>

N S

−

 

_{ }

 

for every

v S

∈

. In this note, we investigate the Majority Independence Number

β

_M

( )

G

for Generalised Petersen graphs and also discussed whether it is

β

_M-excellent or not.

Keywords: Majority independence number-

β

_M

( )

G

,

β

_Mexcellent graphs.

2010 Mathematics Subject Classification: 05C69.

1. INTRODUCTION

We consider connected, undirected, finite graphs without loops. We follow the notations and terminology of Harary[2] and Haynes et al. [3]. Let G=( , )V E be a graph with

V

=

p

and

E

=

q

. For every vertex

v V G

∈

( )

, the open neighbourhood

N v

( )

=

{

u V G

∈

( ) /

uv E G

∈

( )

}

and the closed neighbourhood

N v

[ ]

=

N v

( )



{ }

v

. Let

S

be a set of vertices, and let u∈S. The private neighbor set of

u

with respect to

S

_is

pn u S

[ , ] { /

=

v N v

[ ]



S

=

{ }}

u

In 2006, A subset

S

⊆

V G

( )

of vertices in a graph

G

is called majority dominating set if at least half of the vertices of

( )

V G are either in

S

or adjacent to the vertices of

S

.

i.e.,

[ ]

2

p N S ≥

 

_{ }

 

. A majority dominating set

S

is minimal if no proper subset of

S

is a majority dominating set of

G. The majority domination number

γ

_M

( )

G

of a graph G is the minimum cardinality of a minimal majority dominating set in G. The upper majority domination number _M

( )

G

is the maximum cardinality of a minimal majority dominating set of a graph G. This parameter has been studied by Swaminathan. V and Joseline Manora. J[8].

Corresponding Author: B. John*

1

_,

1

_{Department of Mathematics, A.J.C. English School, Kumbakonam, Tamil Nadu, India.}

M

disconnected subgraph with

[ ]

2

p

N S

≥

 

_{ }

 

and

[ , ]

[ ]

2

p

pn v S

>

N S

−

 

_{ }

 

for every

v S

∈

. If any vertex set

'

S

properly containing S is not majority independent. Then S is called Maximal Majority Independent set. The minimum cardinality of a maximal majority independent set is called lower majority independence number of G and it is also called Independent Majority Domination number of G. It is denoted by

i

_M

( )

G

. The maximum cardinality of a maximal majority independent set of G is called Majority Independence number of G and it is denoted by

β

M

( )

G

. A

M

β

-set is a maximum cardinality of a maximal majority independent set of G. This parameter has been highly developed by Joseline Manora. J and John. B[5].

Claude Berge in 1980, introduced B graphs. These are graphs in which every vertex in the graph is contained in a maximum independent set of the graph. Fircke et al. [1] in 2002 made a beginning of the study of graphs which are excellent with respect to various parameters.

γ

-excellent trees and total domination excellent trees have been studied in [1]. Also in 2006, N.Sridharan and Yamuna [7] made an extensive work in this area. In 2011, Swaminathan. V and Pushpalatha. A.P have defined

β

_ο-excellent graphs, just

β

_ο-excellent graphs and very

β

_ο-excellent graphs and they have made a detailed study in this paper [7].

Definition: For each

n

≥

3

and

0

< <

k n

,

P n k

( )

,

denotes the Generalized Petersen graph with vertex set

{

1 2 1 2

}

( )

,

,...,

_n

,

,

,...,

_n

V G

=

u u

u

v v

v

and the edge set

E G

( )

=

{

u u

_{i i+1 (modn)}

,

u v v v

_{i i}

,

_{i i k+ (modn)}

}

,

1

≤ ≤

i

n

.

Definition: Let

G

=

( , )

V E

be a simple graph. Let

u V G

∈

( )

. The vertex

u

is said to be

β

_M -good if

u

is contained in a

β

_M-set of

G

. The vertex

u

is said to be

β

_M-bad if there exists no

β

_M-set of

G

containing

u

. A graph

G

is said to be

β

_M-excellent if every vertex of

G

is

β

_M-good. This parameter has been studied by Joseline Manora. J and John. B [4].

2. Exact

β

_M-number for

G P n k

=

( , )

Theorem 2.1: Let

G

_{be a Generalization of Petersen graph} with , . Then

Proof: Let be a Generalized Petersen graph with . The graph consists of two cycles and such that the cycle with vertex set is nested by the another cycle with vertex set and each in is joined to exactly one in and

, .

Case-(i): When . The maximum majority independent sets are . Then

, if .

( )

,

P n k

k

=

1

n

≥

3

4

6

4

( )

7

7

3

8

4

M

p

if n

p

G

if n

p

if n

β





−



_≤



_



_



 

=

_{ }

=

 





_

₋

_

≥



_



_



G

P n

( )

,1

V G

( ) 2

= =

n p

G

1

C

C

₂

C

₁

{

v v

₁

,

₂

,...,

v

_n

}

C

₂

{

u u

1

,

2

,...,

u

n

}

u

i

C

2

v

i

C

1

( ) ( )

_i

_i

3

d v

=

d u

=

i

=

1,2,...,

n

6

n

≤

{

v u

i

,

i

+

1 (mod )

n

}

,

i

=

1, 2,...,6

4

( ) 2

4

M

p

G

β

_{= =}



−



_

Case-(ii): When . The maximum majority independent sets are .

Therefore .

Case-(iii): When . Let , and .

Then . Also, for every ,

. Hence is a -set of .

Therefore . Suppose , with

. But , for any . Therefore is not a

-set of . Hence .

⇒

Therefore . The maximal majority independent sets of

G

are

,

.

Proposition 2.2[4]: Let be a Generalization of Petersen graph with

k

=

1,

n

≥

3

. Then

( )

,1

G

=

P n

is -excellent.

Proof: In all the cases of the above theorem [2.1], all vertices of are contained in any one of the -sets of . Therefore all vertices are -good vertices. Hence is -excellent.

Theorem 2.3: Let be a Generalization of Petersen graph with , . Then

Proposition 2.4: Let be a Generalization of Petersen graph with , . Then is -excellent.

7

n

=

{

v

_i

,

u

_i

₊

_{2 (mod )}

_n

,

i

=

1, 2,...,7

}

( ) 2

7

M

p

G

β

_{= =  }

 

 

8

n

≥

D

=

{

u u

₁

,

₂

,..,

u

_t

}

3

4

p

t

=



−



_





d u u

(

i

,

j

)

≥

2 ,

i j

≠

(

)

1

[ ]

( ) 1

t

i i

N D

d u

=

=

∑

+

=

4

t

₄

3

4

2

p

−

p



  

=

_

_{  }

≥



  

v D

∈

[ , ]

pn v D

_{[ ]}

2

p

N D

 

>

_{− }

 

D

β

M

G

3

( )

4

M

p

G

D

β

≥

_{= }



−



_





S

=

{

v v

1

,

2

,..,

v

r

}

3

1

4

p

r

=



_

−



_

+





( )

_i

,

_j

2,

d v v

≥

i

≠

j

pn v S

[ , ]

[ ]

2

p

N S

 

≤

_{− }

 

v S

∈

S

M

β

G

( )

3

1

4

M

p

G

S

β

<

=



_

−



_

+





3

( )

4

M

p

G

β

_{≤ }



−



_





3

( )

4

M

p

G

β

_{= }



−



_





{

v

_i

,

u

_i

₊

_{1(mod )}

_n

,

v

_i

₊

_{2(mod )}

_n

,

u

_i

₊

_{3 (mod )}

_n

,...

}

{

u

_i

,

v

_i

₊

_{1(mod )}

_n

,

u

_i

₊

_{2(mod )}

_n

,

v

_i

₊

_{3 (mod )}

_n

,... ,

}

i

=

1, 2,...,

n

G

P n k

( )

,

M

β

( )

V G

β

_M

G

β

_M

G P n

=

( )

,1

β

_M

G

P n k

( )

,

k

=

2

n

≥

3

11

8

( )

12

6

M

p

if n

G

p

if n

β

 

_≤

 

 

=

 



_≥

 

 



G

P n k

( )

,

k

=

2

n

≥

3

( )

,2

Theorem 2.5: Let be a Generalization of Petersen graph with

k

=

3

, . Then

2

10

6

( )

4

1

11

4

M

p

if n

G

p

if n

β





−



_≤









=

−







₋

_≥











Proof: Let be a Generalized Petersen graph

P n

( ,3)

with vertices. Then consists of two cycles and such that the cycle with vertex set is nested by the another cycle

with vertex set and each in is joined to exactly one in ,

and .

Case-(i): When

n

≤

10

. Let . Then

p

=

2

n

=

10,11,12,13,14

.

Let

D

=

{

u

₁

,

v

₂

}

. . Therefore .

, for

∀ ∈

u

_i

D i

,

=

1, 2

. Therefore is a maximal majority

independent set of

( ) 2

2

6

M

p

G

β



−



⇒

_{= = }

_





.

Let

n

=

8,9,10

. Then

p

=

2

n

=

16,18, 20

. Let

D

=

{

u

₁

,

v u

₂

,

₃

}

.

[ ] 10

2

p

N D

= ≥  

 

 

.

Then

[ , ] 4

3

[ ]

2

i

p

pn u D

=

or

>

N D

−  

 

 

, for

∀ ∈

u

i

D i

,

=

1,3

and

2

[

, ] 3

[ ]

2

p

pn v D

= >

N D

−  

 

 

,

v

2

∈

D

. Therefore

2

( ) 2

6

M

p

G

β

_{= = }



−



_





.

Case-(ii): When

n

≥

11

. Let ,

4

1

4

p

t

=



_

−



_

−





and . Then .

Also, and .

Therefore

[ , ]

[ ]

2

i

p

pn u D

>

N D

−  

 

 

, for . Therefore

D

is a maximal majority independent

set of . Hence

( )

4

1

4

M

p

G

D

β

≥

=



_

−



_

−





.

G

P n k

( )

,

n

≥

3

G

V G

( ) 2

= =

n p

G

1

C

C

₂

C

₁

{

v v

₁

,

₂

,...,

v

_n

}

2

C

{

u u

₁

,

₂

,...,

u

_n

}

u

_i

C

₂

v

_i

C

₁

i

=

1,2,...,

n

( ) ( )

_i

_i

3

d v

=

d u

=

5,6,7

n

=

[ ]

7

2

p

N D

= ≥  

 

 

[ ]

2

2

1

0

p

N D

−

 

_{ }

=

or or

 

[

, ]

3

[ ]

2

i

p

pn u D

= >

N D

− 

 

 

D

G

{

1

,

2

,..,

t

}

D

=

u u

u

(

_i

,

_j

)

2 ,

d u u

≥

i j

≠

1

[ ]

( )

1

t i i

N D

d u

=





=



_



_

+



∑



3

t

1

= +

2

p

 

≥ 

_{ }

[ ]

N D

2

p

 

− 

_{ }

= 



₁

0

_{if n is odd}

if n is even



pn u

[

i

,

D

]

=

4

or

3

or

2

i

u

D

∀ ∈

Suppose ,

4

1 1

4

p

r

=



_

−



_

− +





with .

. But , for any .

is not a -set of . Therefore

( )

4

4

M

p

G

S

β

≤

=



_

−



_





4

( )

1

4

M

p

G

β

−

⇒

≤



_



_

−





.

Hence

( )

4

1

4

M

p

G

β

=



_

−



_

−





.

The maximal majority independent sets of

G P n

=

( ,3)

are

{

,

,

,

;

1, 2,3,...,

}

1(mod )

2 (mod )

3(mod )

u

v

u

v

i

n

i

i

+

n

i

+

n

i

+

n

=

, if

n

=

11,12

,...

{

,

,

,

,

;

1, 2,3,...,

}

1(mod )

2 (mod )

3(mod )

4 (mod )

u

v

u

v

u

i

n

i

i

+

n

i

+

n

i

+

n

i

+

n

=

,

if

n

=

13,14

,...

In general, the maximal majority independent sets of G are

{

,

,

,

;

1, 2,3,...,

}

1(mod )

2(mod )

3(mod )

u

v

u

v

i

n

i

i

+

n

i

+

n

i

+

n

=

,

when

n

=

3

k

−

1,3 ,3

k

k

+

3,3

k

+

4,3

k

+

7,3

k

+

8, ,...,

if

k

=

4

. When

n

=

3

k

+

1,3

k

+

2,3

k

+

5,3

k

+

6,3

k

+

9,3

k

+

10, ...,

if

k

=

4

, then the maximal majority independent sets of G are

{

,

,

,

,

;

1, 2,3,...,

}

1(mod )

2(mod )

3(mod )

4(mod )

u

v

u

v

u

i

n

i

i

+

n

i

+

n

i

+

n

i

+

n

=

.

Proposition 2.6: Let be a Generalization of Petersen graph with

k

=

3,

n

≥

3

. Then

G

=

P n

( )

,3

is -excellent.

Proof: All vertices of are contained in any one of the -sets of by theorem (2.5). Therefore all vertices of

G

=

P n

( )

,3

are -good vertices. Hence

G

=

P n

( )

,3

is -excellent.

CONCLUSION

In this paper we surveyed the -number for the Generalised Petersen graphs

G

=

P n k

( )

,

where

1, 2, 3,

3

k

=

n

≥

and also discussed -excellent. Further we extend this idea to find -excellent and -number for

G

=

P n k

(

,

)

where

k

>

3,

n

≥

3

and also extend this idea for the some more interesting different types of graphs.

REFERENCES

1. Fricke. G.H, Teresa W. Haynes, Hadetniemi .S. T, Hedetniemi. S. M and Laskar. R. C, Excellent trees, Bull. Inst. Combin. Appl. Vol 34(2002), pp 27-38.

{

1

,

2

,..,

r

}

S

=

v v

v

d v v

( ,

_i

_j

)

≥

2 ,

i j

≠

1

[ ]

( )

1

r

i

i

N S

d v

=





=



_



_

+



∑



=

3

r

+

1

2

p

 

≥ 

_{ }

[

, ]

[ ]

2

i

p

pn v S

≤

N S

− 

 

 

v

i

∈

S

S

β

_M

G

G

P n k

( )

,

M

β

( )

V G

β

_M

G

M

β

β

_M

M

β

M

3. Haynes. T. W, Hedetniemi S. T and P. J. Slater, Fundamentals of domination in Graphs, Marcel Dekkar, New York, 1998.

4. Joseline Manora. J, John. B,

β

_M

( )

G

-Excellent Graphs-Proceedings of the International Conference on Jamal Research Journal: (Special Issue) ISSN 0973-0303(2015), pp 513-517.

5. Joseline Manora. J, John. B, Majority Independence Number of a Graph, International Journal of Mathematical Research, Vol-6, No.1 (2014), 65-74.

6. Pushpalatha. A.P, Jothilakshmi. G, Suganthi. S, Swaminathan. V,

β

_ο-excellent graphs, WSEAS Transaction on Mathematics, Issue 2, Vol. 10, February 2011.

7. Sridharan. N and Yamuna. M, A Note on Excellent graphs, ARS COMBINATORIA 78(2006), pp. 267-276. 8. Swaminathan. V, Joseline Manora. J, Majority Dominating Sets of a graph, Jamal Academic Research Journal,

Vol-3, No.2 (2006), 75-82.

Source of support: Nil, Conflict of interest: None Declared.